Georgia Performance Standards Framework for Mathematics Grade 6 Unit Seven Organizer: SCALE FACTOR (3 weeks)

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1 The following instructional plan is part of a GaDOE collection of Unit Frameworks, Performance Tasks, examples of Student Work, and Teacher Commentary. Many more GaDOE approved instructional plans are available by using the Search Standards feature located on GeorgiaStandards.Org. OVERVIEW: Unit Seven Organizer: SCALE FACTOR (3 weeks) In this unit students will: investigate the history of commonly used units of measure; determine appropriate units of measure for length and area; create scale drawings; and use ratios, proportions, and scale factors to compare similar figures and solve problems. To assure that this unit is taught with the appropriate emphasis, depth and rigor, it is important that the tasks listed under Evidence of Learning be reviewed early in the planning process. A variety of resources should be utilized to supplement, but not completely replace, the textbook. Textbooks not only provide much needed content information, but excellent learning activities as well. The tasks in these units illustrate the type of learning activities that should be utilized from a variety of sources. ENDURING UNDERSTANDINGS: It is important to be able to convert from one unit of measure to another within the same system of measurement. Ratios, proportions and scale factors are used to solve many problems encountered in everyday life. The ratios of corresponding sides of similar figures are constant. Scale factors are used to create scale drawings. ESSENTIAL QUESTIONS: Why is it important to be able to convert from one unit of measure to another? How do you determine the most appropriate unit of measure for measuring lengths, areas, and perimeters of simple plane figures? How can scale factors, ratios, and proportions be used to represent the relationships that exist between similar figures? September 20, 2006 Page 1 of 16

2 How can scale factors, ratios, and proportions be used to solve problems related to similar figures? Why do we need scale drawings? STANDARDS ADDRESSED IN THIS UNIT Mathematics standards are interwoven and should be addressed throughout the year in as many different units and activities as possible in order to emphasize the natural connections that exist among mathematical ideas. KEY STANDARDS: M6M1. Students will convert from one unit to another within one system of measurement (customary or metric) by using proportional relationships. M6M2. Students will use appropriate units of measure for finding length, perimeter, area and volume and will express each quantity using the appropriate unit. a. Measure length to the nearest half, fourth, eighth and sixteenth of an inch. b. Select and use units of appropriate size and type to measure length, perimeter, area and volume. c. Compare and contrast units of measure for perimeter, area, and volume. M6G1. Students will further develop their understanding of plane figures. c. Use the concepts of ratio, proportion and scale factor to demonstrate the relationships between similar plane figures. d. Interpret and sketch simple scale drawings. e. Solve problems involving scale drawings. M6A1. Students will understand the concept of ratio and use it to represent quantitative relationships. M6P1. Students will solve problems (using appropriate technology). a. Build new mathematical knowledge through problem solving. b. Solve problems that arise in mathematics and in other contexts. M6P2. Students will reason and evaluate mathematical arguments. a. Recognize reasoning and proof as fundamental aspects of mathematics. M6P3. Students will communicate mathematically. a. Organize and consolidate their mathematical thinking through communication. b. Communicate their mathematical thinking coherently and clearly to peers, teachers, and others. c. Analyze and evaluate the mathematical thinking and strategies of others. d. Use the language of mathematics to express mathematical ideas precisely. September 20, 2006 Page 2 of 16

3 RELATED STANDARDS: M6P1. Students will solve problems (using appropriate technology). c. Apply and adapt a variety of appropriate strategies to solve problems. d. Monitor and reflect on the process of mathematical problem solving. M6P2. Students will reason and evaluate mathematical arguments. b. Make and investigate mathematical conjectures. c. Develop and evaluate mathematical arguments and proofs. d. Select and use various types of reasoning and methods of proof. M6P3. Students will communicate mathematically. d. Use the language of mathematics to express mathematical ideas precisely. M6P4. Students will make connections among mathematical ideas and to other disciplines. a. Recognize and use connections among mathematical ideas. b. Understand how mathematical ideas interconnect and build on one another to produce a coherent whole. c. Recognize and apply mathematics in contexts outside of mathematics. M6P5. Students will represent mathematics in multiple ways. a. Create and use representations to organize, record, and communicate mathematical ideas. b. Select, apply, and translate among mathematical representations to solve problems. c. Use representations to model and interpret physical, social, and mathematical phenomena. CONCEPTS/SKILLS TO MAINTAIN: It is expected that students will have prior knowledge/experience related to the concepts and skills identified below. It may be necessary to pre-assess in order to determine if time needs to be spent on conceptual activities that help students develop a deeper understanding of these ideas. estimate the area of fundamental geometric plane figures model multiplication and division of decimals September 20, 2006 Page 3 of 16

4 SELECTED TERMS AND SYMBOLS: The following terms and symbols are often misunderstood. These concepts are not an inclusive list and should not be taught in isolation. However, due to evidence of frequent difficulty and misunderstanding associated with these concepts, instructors should pay particular attention to them and to how their students are able to explain and apply them. Proportion: An equation which states that two ratios are equal. Ratio : A comparison of two quantities that have the same unit of measure. Scale Drawings : Drawings that represent relative sizes and placements of real objects or places. Scale Factor : The ratio of corresponding lengths of the sides of two similar figures. Similar Figures : Figures that have the same shape but not necessarily the same size. You may visit and click on dictionary to see definitions and specific examples of terms and symbols used in the sixth grade GPS. EVIDENCE OF LEARNING: By the conclusion of this unit, students should be able to demonstrate the following competencies: Convert from one unit of measure to another within the same system of measurement; Measure lengths to nearest ½, ¼ and 1/16 of an inch; Select and use units of appropriate size and type to measure length and area; Use scale factors, ratios, and proportions to describe relationships between similar figures; Use scale factors, ratios, and proportions to solve problems involving similar figures; and Create scale drawings. The following task represents the level of depth, rigor, and complexity expected of all 6 th grade students. This task or a task of similar depth and rigor should be used to demonstrate evidence of learning. Culminating Activity: Designer s Challenge An activity where students create plans and blueprints for redecorating/rearranging a room in their house. September 20, 2006 Page 4 of 16

5 STRATEGIES FOR TEACHING AND LEARNING: Students should be actively engaged in developing their own understanding. Mathematics should be represented in as many ways as possible by using graphs, tables, pictures, symbols and words. Appropriate manipulatives and technology should be used to enhance student learning. Students should be given opportunities to revise their work based on teacher feedback, peer feedback, and metacognition which includes self-assessment and reflection. TASKS: The collection of the following tasks represents the level of depth, rigor and complexity expected of all sixth grade students to demonstrate evidence of learning. The History of Measurement The History of Measurement Have students investigate how different units of measure came to be by using the Internet or the school library. ********************************************************************************************* The History of Measurement Discussion, Suggestions, Possible Solutions This introduction to history of measurement should provoke discussion about how units that we commonly use came to be. Students may investigate this individually or in pairs in the computer lab, at the library or as a homework assignment. Students should report their findings to the class. The teacher will need to specify expectations for the report. Solutions will vary. September 20, 2006 Page 5 of 16

6 Suggested children s book How Big is a Foot? by Rolf Myller. This book humorously shows the problems with nonstandard forms of measurement. Suggested web sites: The Metric System This web site gives a brief overview of the Metric System, how it came to be, and why. The United States and the Metric System This web site discusses the history of the metric system as it relates to the United States. International System of Units from NIST History of Mathematics: History of Numerals and Counting A bibliography of various books about the history of numbers and counting. U.S customary units Facts, Info, and Encyclopedia article: This web site gives a brief overview of the customary units. Math: Customary or Standard Measurement ethemes emints There are numerous other web sites that house information about the customary system. Go to each of the web sites and determine their appropriateness before sending students to a site. September 20, 2006 Page 6 of 16

7 New Unit New Unit a) Create a unit of length so that the pencil above would be more than 2 units long but less than three units long. Explain how you decided on the length of your unit. b) Name your new unit and provide conversions for the metric and customary systems. c) In your small group, each person is also creating their own unit of length. Measure the lengths of several items within the classroom using your new units. Compare your results with the results of your group members. d) Do you think it is necessary to have standard units of measure? Why or why not? Explain your thinking. ********************************************************************************************* New Unit Discussion, Suggestions, Possible Solutions This task is adapted from the website Open-Ended Assessment in Math: Students may work on this task for finding a new unit individually and then report to small group or whole class. Allow the students to use calculators if necessary. Solutions will vary based on the length of the pencil viewed either in print or through other means. Possible solution: My pencil length is 10.7 cm or 4 and 3/16 inches long. Since my pencil needs to be more than two units long, my new unit needs to be less than half the length of my pencil. Since the pencil needs to be less than 3 new units long, my new unit needs to be more than 1/3 the length of the pencil. Half the length of the pencil is 5.35 centimeters. One-third the length of my pencil is centimeters. My new unit must be between and 5.35 centimeters in length. b. My new unit of measure is called a gorp and it measures 5 cm long. A gorp is approximately 2 inches. Therefore, my pencil is 2.14 gorps in length. c. and d. Answers will vary. September 20, 2006 Page 7 of 16

8 A Common Misconception About Scaling A Common Misconception About Scaling Johnny is working on the following problem: The picture on a poster that is 4 feet wide and 6 feet long is to be scaled down and drawn onto a small poster that is 1 foot wide. How long should the small poster be? Johnny solves the problem this way: One foot is 3 feet less than 4 feet, so the length of the small poster should also be 3 feet less than the length of the big poster. This means the small poster should be 6 3 = 3 feet long. Is Johnny s reasoning valid? Why or why not? If not, how might you convince Johnny that his reasoning is not correct? What would be a correct way to solve the problem in that case? ********************************************************************************************* A Common Misconception About Scaling Discussion, Suggestions, Possible Solutions The resource for this task is Beckmann, S. (2005). Class activities to accompany mathematics for elementary teachers. Addison Wesley: Boston Students may work on the activity in pairs or in small groups. Johnny s reasoning is not valid because the ratio of his poster side lengths is not the same for the original and for this smaller poster. The original poster is 4 feet wide and 6 feet long, which allows for a width to length ratio of 4 to 6, which reduces to 2 to 3. The smaller poster is now 1 foot wide and 3 feet long, which has a width to length ratio of 1 to 3, which is not the same as the ratio for the original poster. In order to maintain the original shape of the poster, the ratios would have to stay the same. September 20, 2006 Page 8 of 16

9 Another way to look at the ratios includes looking at the ratio of the widths of the original poster to the smaller poster, which is 4 to 1. The ratio of the lengths of the original poster to the smaller poster should be the same as this. The lengths ratio is 6 to 3, which reduces to 2 to 1. Again, the ratios are not the same. We could ask Johnny to draw the dimensions of his new smaller poster to see if it looks similar to the original poster. Hopefully, Johnny would see that the rectangles do not have the same shape in terms of their dimensions. We could then show Johnny several different proportions that would yield the correct length for the smaller poster. For example: Using the width to length ratio of the original poster: Using the original poster width to the smaller poster width: width length 4 = 6 1 x 4x = 6 original smaller 4 = 1 6 x 4x = 6 x = 6 4 x = 6 4 x = x = When using either ratio, we get that the length of the smaller poster should be 1.5 feet, not 3 feet! Similar Figures September 20, 2006 Page 9 of 16

10 Similar Figures a) Triangle ECD is similar to triangle EAB. What is the scale factor for the two triangles? Explain how you found the scale factor. Find side AB. Find the perimeters of triangle ECD and triangle EAB. What is the ratio of the perimeters? Is it the same or different from the scale factor? Why do you think it is the same or different? b) Gregory is in the sixth grade and is working on a project having to do with agricultural use of land in South Georgia. He has a small copy of a map of a farm in his area but needs to make a much larger version on which he can record all accumulated data. The map he wants to enlarge is shown below. He has decided that he wants the length of the northern boundary of the farm for the larger version to be 24 inches. How would you explain to him what he should do to find the lengths of the other sides so that the enlarged map is a scale drawing of the smaller one? Find the area and perimeter of the original map of the farm and the larger map. Find the ratios of the areas and of the perimeters. How do these ratios relate to the scale factor of the maps? Are they the same ratios or different? Why do you think they are the same or different? September 20, 2006 Page 10 of 16

11 c) Explain the following relationships: the ratios between the sides of two similar figurs the ratios between the perimeters of two similar figures the ratios between the areas of two similar figures ********************************************************************************************* Similar Figures Discussion, Suggestions, Possible Solutions: Students may work on this activity individually or in pairs. a.) The scale factor of the two triangles is 3:4, which was found by using the length of side EC and the length of corresponding side EA as the ratio. Using this ratio, we can find the length of side AB: EC CD 3 4 = = EA AB 4 x 3x = 16 x = 16 3 x = September 20, 2006 Page 11 of 16

12 Therefore, the length of AB is 5 1/3 feet. A common misconception is that the ratio of the triangles would involve EC and CA; however, CA is not the side of the triangle! The perimeter of triangle ECD is 10 ft and the perimeter of triangle EAB is 13 1/3 ft. The ratio of the perimeters is 10:13 1/3, which does not obviously match the ratio of the side lengths at 3:4. However, this perimeter ratio does reduce to 3:4. There are three 3 1/3 in 10 and there are four 3 1/3 in 13 1/3. So, if we reduce the perimeter ratio by a factor of 3 1/3, we get the side lengths ratio of 3:4. Therefore, the side length ratio is maintained with perimeter. b.) We could explain to Gregory that he can use the ratio that he created to set up proportions with each side of the farm map to find those other lengths. For example, his ratio is 16:24, which will reduce to 2:3. He can use the ratio of 2:3 to find the length of the side corresponding to the side length of 12 inches on his smaller map this way: 16 2 = 24 3 so the proportion could be written as = x 2x = 36 x = 36 2 x = 18 So the larger map would have a side length of 18 inches corresponding to the side on the smaller map with a side length of 12 inches. This ratio could be used over and over to calculate all the new side lengths. Another method that might make sense to Gregory is a little less procedural. 16 is to 24 where 24 = 1/ = Proportional reasoning is still used even though it does not appear as a proportion. The perimeter of the original map can be found by adding all the values of the side lengths: = 56 in The perimeter of the larger map can be found by adding the corresponding side lengths: = 84 in The perimeters are also in the ratio of 2:3 just like the side lengths. September 20, 2006 Page 12 of 16

13 The area of the original map can be found by breaking the figure into smaller rectangles and computing each area: 168 in 2 The area of the larger map can be found using the same method: 378 in 2 The areas are in the ratio of 4:9, which is actually the square of the original ratio. Designer s Challenge This culminating task represents the level of depth, rigor and complexity expected of all 6 th grade students to demonstrate evidence of learning. UNIT SEVEN TASK: Designer s Challenge a) Create a scale drawing of a room in your house. b) Decide on a redecorating plan. (Be creative!) b) Create an expense sheet including headings for all monies needed for redecorating such as paint, wall paper, border, material for covering pillows, draperies, furniture, etc. c) Your teacher will give you a budget. Determine the total amount of money your plan will cost. You will need to research prices and then compute amounts of material that you will need by finding areas of walls for painting or wallpapering, perimeter for borders, areas for pillow covers and drapes, etc. d) Create a poster in order to present your design to the class. KEY STANDARDS: Standards Addressed in this Task M6M1. Students will convert from one unit to another within one system of measurement (customary or metric) by using proportional relationships. September 20, 2006 Page 13 of 16

14 M6M2. Students will use appropriate units of measure for finding length, perimeter, area and volume and will express each quantity using the appropriate unit. a. Measure length to the nearest half, fourth, eighth and sixteenth of an inch. b. Select and use units of appropriate size and type to measure length, perimeter, area and volume. c. Compare and contrast units of measure for perimeter, area, and volume. M6G1. Students will further develop their understanding of plane figures. c. Use the concepts of ratio, proportion and scale factor to demonstrate the relationships between similar plane figures. d. Interpret and sketch simple scale drawings. e. Solve problems involving scale drawings. M6P1. Students will solve problems (using appropriate technology). a. Build new mathematical knowledge through problem solving. b. Solve problems that arise in mathematics and in other contexts. M6P3. Students will communicate mathematically. a. Organize and consolidate their mathematical thinking through communication. b. Communicate their mathematical thinking coherently and clearly to peers, teachers, and others. RELATED STANDARDS: M6A1. Students will understand the concept of ratio and use it to represent quantitative relationships. M6P1. Students will solve problems (using appropriate technology). c. Apply and adapt a variety of appropriate strategies to solve problems. d. Monitor and reflect on the process of mathematical problem solving. M6P2. Students will reason and evaluate mathematical arguments. a. Recognize reasoning and proof as fundamental aspects of mathematics. b. Make and investigate mathematical conjectures. c. Develop and evaluate mathematical arguments and proofs. d. Select and use various types of reasoning and methods of proof. September 20, 2006 Page 14 of 16

15 Whole numbers and decimal fraction computations Length, perimeter, and area Concepts/Skills to Maintain Suggestions for Classroom Use While this task may serve as a summative assessment, it also may be used for teaching and learning. It is important that all elements of the task be addressed throughout the learning process so that students understand what is expected of them. The task can be given to the students at the beginning of the unit. As they are working on the Supporting tasks, they can be thinking about this project and using instructional time appropriately. As an example, the culminating task can be organized into parts to aid students in their tasks: 1.) Draw scale drawing of chosen room 2.) Students are given a budget for materials, such as paint, wallpaper, borders, fabric for pillows, etc. 3.) Students create a few redesign options students should be able to explain their designs. 4.) Students calculate which design options fall within their given budget. 5.) Students create a poster to present their design to the class, which includes their design, budget, materials needed, etc. Technology can be easily integrated into this lesson by having the students use a spreadsheet program to work on their budgets. After they have determined what materials they need and how much it will cost, they can then create a circle graph to represent what percentage of their budget goes to which items for their final report. The teacher might also want to have an interior designer come in and speak with the class to talk about concerns that involve redesigning a room such as knowing who the customer is, how the room is going to be used, etc. September 20, 2006 Page 15 of 16

16 Discussion, Suggestions and Possible Solutions While this task may serve as a summative assessment, it also may be used for teaching and learning. It is important that all elements of the task be addressed throughout the learning process so that students understand what is expected of them. The task can be given to the students at the beginning of the unit. As they are working on the Supporting tasks, they can be thinking about this project and using instructional time appropriately. As an example, the culminating task can be organized into parts to aid students in their tasks: 6.) Draw scale drawing of chosen room 7.) Students are given a budget for materials, such as paint, wallpaper, borders, fabric for pillows, etc. 8.) Students create a few design options students should be able to explain their designs. 9.) Students calculate which design options fall within their given budget. 10.)Students create a poster to present their design to the class, which includes their design, budget, materials needed, etc. Students may vote on the best 3 to be displayed in the classroom or in the hall for the rest of the quarter. A possible extension of the task could be to redesign the classroom. Technology can be easily integrated into this lesson by having the students use a spreadsheet program to work on their budgets. After they have determined what materials they need and how much it will cost, they can then create a circle graph to represent what percentage of their budget goes to which items for their final report. The teacher might also want to have an interior designer come in and speak with the class to talk about the mathematics involved in redesigning a room. September 20, 2006 Page 16 of 16

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